Solve 500x^2+496x+186=0

$$500 { x }^{ 2 } +496x+186 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{7874}i}{250}-\frac{62}{125}\approx -0.496+0.354942249i$$

$$x=-\frac{\sqrt{7874}i}{250}-\frac{62}{125}\approx -0.496-0.354942249i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$500x^{2}+496x+186=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 500 for a, 496 for b, and 186 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-496±\sqrt{496^{2}-4\times 500\times 186}}{2\times 500}$$

Square 496.

$$x=\frac{-496±\sqrt{246016-4\times 500\times 186}}{2\times 500}$$

Multiply -4 times 500.

$$x=\frac{-496±\sqrt{246016-2000\times 186}}{2\times 500}$$

Multiply -2000 times 186.

$$x=\frac{-496±\sqrt{246016-372000}}{2\times 500}$$

Add 246016 to -372000.

$$x=\frac{-496±\sqrt{-125984}}{2\times 500}$$

Take the square root of -125984.

$$x=\frac{-496±4\sqrt{7874}i}{2\times 500}$$

Multiply 2 times 500.

$$x=\frac{-496±4\sqrt{7874}i}{1000}$$

Now solve the equation x=\frac{-496±4\sqrt{7874}i}{1000} when ± is plus. Add -496 to 4i\sqrt{7874}.

$$x=\frac{-496+4\sqrt{7874}i}{1000}$$

Divide -496+4i\sqrt{7874} by 1000.

$$x=\frac{\sqrt{7874}i}{250}-\frac{62}{125}$$

Now solve the equation x=\frac{-496±4\sqrt{7874}i}{1000} when ± is minus. Subtract 4i\sqrt{7874} from -496.

$$x=\frac{-4\sqrt{7874}i-496}{1000}$$

Divide -496-4i\sqrt{7874} by 1000.

$$x=-\frac{\sqrt{7874}i}{250}-\frac{62}{125}$$

The equation is now solved.

$$x=\frac{\sqrt{7874}i}{250}-\frac{62}{125} x=-\frac{\sqrt{7874}i}{250}-\frac{62}{125}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$500x^{2}+496x+186=0$$

Subtract 186 from both sides of the equation.

$$500x^{2}+496x+186-186=-186$$

Subtracting 186 from itself leaves 0.

$$500x^{2}+496x=-186$$

Divide both sides by 500.

$$\frac{500x^{2}+496x}{500}=-\frac{186}{500}$$

Dividing by 500 undoes the multiplication by 500.

$$x^{2}+\frac{496}{500}x=-\frac{186}{500}$$

Reduce the fraction \frac{496}{500} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{124}{125}x=-\frac{186}{500}$$

Reduce the fraction \frac{-186}{500} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{124}{125}x=-\frac{93}{250}$$

Divide \frac{124}{125}, the coefficient of the x term, by 2 to get \frac{62}{125}. Then add the square of \frac{62}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{124}{125}x+\left(\frac{62}{125}\right)^{2}=-\frac{93}{250}+\left(\frac{62}{125}\right)^{2}$$

Square \frac{62}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{124}{125}x+\frac{3844}{15625}=-\frac{93}{250}+\frac{3844}{15625}$$

Add -\frac{93}{250} to \frac{3844}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{124}{125}x+\frac{3844}{15625}=-\frac{3937}{31250}$$

Factor x^{2}+\frac{124}{125}x+\frac{3844}{15625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{62}{125}\right)^{2}=-\frac{3937}{31250}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{62}{125}\right)^{2}}=\sqrt{-\frac{3937}{31250}}$$

Simplify.

$$x+\frac{62}{125}=\frac{\sqrt{7874}i}{250} x+\frac{62}{125}=-\frac{\sqrt{7874}i}{250}$$

Subtract \frac{62}{125} from both sides of the equation.

$$x=\frac{\sqrt{7874}i}{250}-\frac{62}{125} x=-\frac{\sqrt{7874}i}{250}-\frac{62}{125}$$